Keywords: semiconductor, quantum wire, spin-orbit coupling, spintronic, perturbation theory, finite element method.

In this paper a detailed analysis is presented of a quasi one-dimensional parabolic nanowire formed in the plane of a two dimensional electron gas (2DEG). The nanowire is modulated by a longitudinal periodic potential which has a form of truncated Fourier co-sines. The Hamiltonian of the wire is rotated in such a way that the non-diagonal part of the total Hamiltonian can be treated as a perturbation. Based on the second order perturbation theory, the energy spectrum of the electron spin states of a III-V semiconductor nanowire was found. The analytically obtained energy spectrum was compared to those values obtained from a numerical method. For numerical simulations, the corresponding eigenvalue problem based on the finite element method was solved.

It was also found that the energy spectrum curves of the electron with opposite spins differed by one quantum number (i.e., the first and second excited states) cross each other with the accessible values of the Rashba-Dresselhaus spin-orbit coupling. The crossing point was investigated using both theoretical and numerical methods. Next, it was confirmed that the crossing point can be manipulated towards the Gamma or L points with the application of Rashba-Dresselhaus spin-orbit coupling. It was demonstrated that the crossing point can be found for relatively larger values of k in GaAs nanowires compared to the corresponding values in InAs nanowires. It is concluded that in GaAs nanowires, the spin splitting energy is mainly dominated by the Dresselhaus spin-orbit coupling, whereas in InAs nanowires, the spin splitting energy is mainly dominated by the Rashba spin-orbit coupling.

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